This section quantifies aliasing in the general case. This result is
then used in the proof of the sampling theorem in the next section.

It is well known that when a continuous-time signal contains energy at
a frequency higher than half the sampling rate, sampling
at samples per second causes that energy to alias to a
lower frequency. If we write the original frequency as
, then the new aliased frequency is
,
for
. This phenomenon is also called ``folding'',
since is a ``mirror image'' of about . As we will
see, however, this is not a complete description of aliasing, as it
only applies to real signals. For general (complex) signals, it is
better to regard the aliasing due to sampling as a summation over all
spectral ``blocks'' of width .

Then the spectrum of the sampled signal is related to the
spectrum of the original continuous-time signal by

The terms in the above sum for are called aliasing
terms. They are said to alias into the base band. Note that the summation of a spectrum with
aliasing components involves addition of complex numbers; therefore,
aliasing components can be removed only if both their amplitude
and phase are known.

Proof:
Writing as an inverse FT gives

Writing as an inverse DTFT gives

where
denotes the normalized discrete-time
frequency variable.

The inverse FT can be broken up into a sum of finite integrals, each of length
, as follows:

Let us now sample this representation for at to obtain
, and we have

since and are integers.
Normalizing frequency as
yields

Since this is formally the inverse DTFT of
written in terms of
,
the result follows.